Recurrences
Derivatives
\begin{align*}
J'_{\nu +1} (x) &= \frac{1}{2} \left[ J_{\nu -1} (x) - J_{\nu +1} (x) \right] ,
\\
J'_{\nu} (x) &= J_{\nu -1} (x) - \frac{\nu}{x}\, J_{\nu} (x) ,
\\
J'_{\nu} (x) &= \frac{\nu}{x}\, J_{\nu} (x) - J_{\nu +1} (x) ,
\\
\frac{\text d}{{\text d}x} \left[ x^{\nu} J_{\nu} (x) \right] &= x^{\nu} J_{\nu -1} (x) ,
\\
\frac{\text d}{{\text d}x} \left[ x^{-\nu} J_{\nu} (x) \right] &= -x^{-\nu} J_{\nu +1} (x) .
\end{align*}
Recurrence Relations for Bessel functions of the first kind
\begin{align*}
J_{\nu +1} (x) &= \frac{2\nu}{x}\, J_{\nu} (x) - j_{\nu -1} (x) ,
\end{align*}
Recurrence Relations for Bessel functions of the second kind
Bessel functions of the second kind, being solutions of the Bessel equation, satisfy the same recurrence relations as the Bessel functions of the first kind. Specifically,
\begin{align*}
Y_{\nu -1} (x) - Y_{\nu +1} (x) &= 2\,Y_{\nu} (x) ,
\\
Y_{\nu -1} (x) + Y_{\nu +1} (x) &= \frac{2\nu}{x}\, Y_{\nu} (x) .
\end{align*}
We also have the relation
\[
Y_{-n} (x) = (-1)^n Y_n (x) ,
\]
when n is an integer.
- Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4.
- Dutka, J., On the early history of Bessel functions, Archive for History of Exact Sciences, volume 49, pages 105–134 (1995). https://doi.org/10.1007/BF00376544
- Watson, G.N., A Treatise on the Theory of Bessel Functions,
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